(1)This is the peer reviewed version of the following article: Cano-Rodríguez, M

This is the peer reviewed version of the following article:

Cano-Rodríguez, M. and Núñez-Nickel, M. (2015), Aggregation Bias in Estimates of

Conditional Conservatism: Theory and Evidence. Journal of Business Finance & Accounting, 42: 51-78,

which has been published in final form at https://doi.org/10.1111/jbfa.12099.

This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited."

AGGREGATION BIAS IN ESTIMATES OF CONDITIONAL CONSERVATISM:

THEORY AND EVIDENCE Abstract

This paper documents a study about the influence of the aggregation effect on the

estimates of models based on the original Basu model – specifically the Ball, Kothari, and Nikolaev model (Ball et al., 2013b). We provide an analytical study of the effect, showing that it can produce two biases: an omitted-variable bias and a truncated-sample bias.

Using separate proxies for good and bad news for each company and year, we estimate the empirical sign and magnitude of those biases. Our results show that the estimates of conditional conservatism based on regressions of (unexpected) earnings on (unexpected) returns, as in Ball et al. (2013b), are contaminated by substantial aggregation bias. More specifically, the aggregation effect causes these models to underestimate good-news timeliness and overestimate bad-news timeliness, thereby overestimating differential timeliness. Moreover, when we use proxies that provide better control for the aggregation effect, the differential timeliness coefficient tends to 0, showing that the influence of conditional conservatism on the returns–earnings relationship is at best marginal.

Keywords: accounting conservatism; conditional conservatism; returns–earnings relationship; aggregation effect.

AGGREGATION BIAS IN ESTIMATES OF CONDITIONAL CONSERVATISM: THEORY AND EVIDENCE

I. INTRODUCTION.

Although the Basu model (Basu, 1997) has become the prevalent model for testing the existence of conditional conservatism (Ball et al., 2013a; Khan and Watts, 2009; Ryan, 2006), it presents various weaknesses that cast doubt on its reliability (for instance, Dietrich et al., 2007; Givoly et al., 2007). In particular, Patatoukas and Thomas (2011) demonstrate the existence of a scale effect that could bias the estimates of the Basu model.

Ball, Kothari, and Nikolaev (2013b) (BKN henceforth) argue that the scale effect can be produced by the relationship between expected earnings and expected returns, and they propose a modified version of the Basu model based on the relationship between unexpected earnings and unexpected returns. Patatoukas and Thomas (2013), have analyzed the time-series and cross-sectional estimates of the BKN model variations, however, noting that the variation observed for BKN estimates is too wide compared with the actual variation in conditional conservatism. Given that the BKN estimates are supposed to measure conservatism, and that conservatism is not expected to present such high variation, they conclude that the BKN estimates are not fully free of bias.

This paper contributes to the literature by showing that the BKN model estimates are highly likely to be biased because of the aggregation effect. This aggregation effect was initially revealed at the empirical level by Givoly et al. (2007), and it affects both the BKN and the original Basu models, because these models use the aggregated market returns of the period rather than separating good news from bad news in the period of

consideration. Although Givoly et al. (2007) concluded that the aggregation effect has a dampening effect on the Basu measure of conservatism, our analysis reveals that it has a more complex influence, producing two types of bias in the estimates of the BKN model:

1) an omitted-variable bias, which arises because the BKN model does not control for the influence of good (bad) news on unexpected earnings when estimating the relationship between unexpected earnings and positive (negative) unexpected returns; and 2) a truncated-sample bias, which arises because the timeliness coefficient for good (bad) news in the BKN model is not estimated using the full sample, but only those

observations with positive (negative) unexpected returns.

Using separate proxies of good and bad news, we show that both good- and bad- news timeliness coefficients are higher for the subsample of observations with negative unexpected returns than for the subsample of observations with positive unexpected returns. This asymmetry introduces a significant truncated-sample bias in the Basu and BKN models, leading them to underestimate good-news timeliness and to overestimate bad-news timeliness, thereby overestimating differential timeliness. Additionally, differential timeliness moves closer to 0 when we use empirical proxies that introduce better control for the aggregation bias, showing that the influence of conditional conservatism on the returns–earnings relationship is, at best, marginal.

In structuring the remainder of the paper, we first review the Basu and BKN models. In Section 3, we show analytically how these models are affected by the

aggregation effect and describe the two types of biases that arise. In Section 4, we define our empirical proxies for good and bad news and use them to estimate the sign and

magnitude of the aggregation biases, using archival data. Finally, we draw our conclusions in Section 5.

II. THE DIFFERENTIAL TIMELINESS COEFFICIENT AS A MEASURE OF CONDITIONAL CONSERVATISM

In his seminal paper, Basu (1997) defines conservatism as the requirement of a higher degree of verification for the accounting recognition of good news than for bad news.

According to this definition, earnings would respond, on average, more quickly to bad news than to good news. Basu’s measure of conservatism is then based upon two conditional relationships (Dietrich et al., 2007; Patatoukas and Thomas, 2013):

X_it

M_it−1 = a⁺+ b⁺⋅ R_it+ ξ_it⁺

|

^{Rit≥0 , and}

−¿

−¿⋅R_it+ ξ_it^¿ X_it ¿

M_it−1 = a^{−¿+ b¿Rit<0,}

¿

(1)

where Xit represents accounting earnings, Mit−1 symbolizes the market value of equity at the beginning of the period, and Rit represents stock returns. Basu’s measure of

conservatism is the difference between the coefficients b⁻ and b⁺, commonly addressed as differential timeliness¹. To estimate differential timeliness, Basu united the two former conditional models into a single piecewise model:

X_it

M_it−1= a₀+ a₁⋅ d_it+ b₀⋅ R_it+ b₁⋅d_it⋅R_it+ξ_it, (2) in which dit is a dummy variable that takes the value of 1 if Rit is negative and 0 otherwise.

Regarding the coefficients, a0 is equal to the coefficient a⁺ and b0 to the coefficient b⁺, a1

1 Although Basu (1997) proposed other measures of conservatism, the differential timeliness is the most widely used in the previous literature.

= a⁻ − a⁺,and b1 = b⁻ − b⁺. Basu (1997) and numerous researchers after him have reported a positive value for the asymmetric timeliness coefficient b1, which has been interpreted as bad news being recognized more quickly than good news and, consequently, as evidence of accounting conservatism.

Despite the popularity of the Basu model, various authors have questioned its reliability (Dietrich et al., 2007; Gigler and Hemmler, 2001; Givoly et al., 2007). In particular, Patatoukas and Thomas (2011) argue that stock price is negatively related to both stock-returns variance and deflated accounting earnings. These negative correlations can generate a positive differential timeliness coefficient in the Basu model, even in the absence of conditional conservatism.

Ball, Kothari, and Nikolaev (2013b) argue that the former scale problem is produced primarily by the cross-sectional relationship between expected returns and expected earnings. They propose a modified version of −the standard Basu model (the BKN model), replacing earnings and returns with unexpected earnings and unexpected returns. The conditional equations of the BKN model can be expressed as

U X_it

M_it−1 = a⁺+ b⁺⋅ U R_it+ ξ_it⁺

|

^{U Rit≥0, and}

−¿

−¿⋅U R_it+ ξ_it^¿ U X_it ¿

M_it−1 =a^{−¿+ b¿U Rit<0,}

¿

(3)

where UXit is unexpected earnings and URit represents unexpected returns. The piecewise form of the model would be:

U X_it

M_it−1= a₀+ a₁⋅ d_it+ b₀⋅ U R_it+ b₁⋅d_it⋅ U R_it+ξ_it, (4)

Using this model, BKN show that, although the magnitude of the differential timeliness coefficient is lower than that observed for the standard Basu model, it is still positive and significant, demonstrating the existence of conditional conservatism even after controlling the scale problem.

Patatoukas and Thomas (2013) show, however, that the BKN estimates exhibit too wide a time-series and cross-sectional variation. Furthermore, they also argue that various recent papers (Banker et al., 2013; Beaver and Ryan, 2009; Collins et al., 2012; Hsu et al., 2012) indicate that the observed asymmetry in the recognition of good and bad news can originate from causes other than conservatism.

III. THE AGGREGATION EFFECT AND ITS INFLUENCE ON BKN ESTIMATES

The first work to address the aggregation effect in the Basu model was that of Givoly et al. (2007). They note that the Basu model does not capture the influence of each

individual economic event on earnings, but rather depicts the relationship between the aggregation of all the economic events occurring during a period and the accumulated effect of all these events on earnings. Using simulated data, they show that the earnings–

returns relationship observed for the aggregated data may be different from the relationship for the individual data.

In this section, we conduct an in-depth analysis of the influence of the aggregation effect on the estimates of the BKN model.

The influence of good and bad news on market returns

On the assumption that capital markets are semi-strong efficient, we can expect market prices to reflect all the publicly available information in a timely fashion (Ball et al.,

2013a). Consequently, the economic effect of any unexpected event will be instantaneously incorporated into the market value of equity.

The market value of equity at any given moment t can be represented by the expression

M_it= E_{t −1}

[

^Mit

]

⁺

∑

j=1 nit

Δ m_{j, it}= E_{t −1}

[

^Mit

]

^{+Δ M}it, (5)

where Mit is a random variable that symbolizes the market value of equity of firm i at the end of period t, Et−1[.] is the expectation operator at the beginning of period t, nit is the number of economic events that produce an unexpected alteration in the market value of equity between t−1 and t, Δmj,it is the value of that unexpected variation in the market value produced by the event j², and ΔMit is the sum of all those unexpected variations.

We now consider that some of the unexpected events of the period constitute good news (they increase the market value of equity) and the remaining unexpected events are bad news (they decrease the market value of equity). We denote positive (negative) variations in the market value of equity produced by good news (bad news) as Δm⁺j,it (Δm⁻j,it). Thus, Δm⁺j,it (Δm⁻j,it) will be equal to Δmj,it if Δmj,it is positive (negative) and 0 otherwise. ΔMit

can then be expressed as

−¿, Δm^−¿_{j ,it}=ΔM_it⁺+Δ M^¿_it ΔM_it=

∑

j=1 nit

Δ m⁺_{j, it}+

∑

j =1 nit

¿

(6)

being ΔM−¿_it^¿ ΔM_it⁺¿

the sum of all the unexpected positive (negative) variations in Mit. By

dividing the variation of the market value during the period t by the market value of

2 We assume that Δmj,it represents stationary and symmetric random variables.

equity at the beginning of the period and substituting Mit by its value according to (5) and (6), we find that market returns can be calculated as

E_{t −1}

[

^Mit

]

^{+Δ M}it

++ΔM^−¿−M_it ^{it −1}

M_it−1 = ΔM^−¿_it

M_it−1 =E_t−1

[

^Rit

]

^{+ U R}it

++ U R_it^{−¿=Et−1[Rit]+ URit,}

¿ R_it=M_it−M_it−1

M_it−1 =¿=Et −1

[

^Mit

]

^−Mit−1

M_it−1 +Δ M⁺_it M_it−1 +¿

(7)

where Rit is the market return of period t;

U R−¿_it^¿ U R_it⁺¿

is the sum of all positive (negative)

unexpected variations in the market value of equity occurring during period t, divided by the market value of equity at the beginning of the period; and URit is the sum of all those unexpected variations, whether positive or negative.

The influence of good news and bad news on accounting net income

According to BKN, we differentiate accounting net income into two parts: the expected accounting net income and the unexpected part of net income:

X_it= E_t−1

[

^Xit

]

^{+ U X}it. (8)

Given that both U R_it⁺ and −¿

U R_it^¿ are unexpected, their effect on accounting earnings will occur through UXit. In this model, however, we assume that accounting net income does not reflect the unexpected economic gains and losses in as timely a fashion as the market value does. This assumption is based on the fact that accounting gains or losses are not recorded unless they meet the accounting recognition criteria based on requirements of verifiability, objectivity, and conservatism.

Under accounting conservatism, losses are recorded in a more timely fashion than gains are (Basu, 1997). We incorporate this possibility by differentiating between the recognition of economic gains and losses. We therefore express unexpected accounting net income as

−¿⋅Δ m^−¿_{j, it}, β^¿_{j, it}

U Xit=

∑

j =1 n_it

β+j ,it⋅Δm+j ,it

+

∑

j=1 n_it

¿

(9)

where β⁺j,it indicates the portion of the unexpected economic gain Δm⁺j,it that meets the requirements to be registered as accounting gains during period t; analogously, β⁻j,it

indicates the portion of the unexpected economic loss Δm⁻j,it that meets the requirements to be registered as accounting losses during the same period. We assume that β⁺j,it and β⁻j,it

are stationary and symmetric random variables, with means ´β⁺ and ^β

−¿

¿´ , respectively. We also assume that they are independent of the economic gain or loss (Δm⁺j,it or Δm⁻j,it)³.

To make our model comparable to the BKN model (equation (4)), we divide the two parts of the equation by the market value at the beginning of the period:

−¿⋅Δm^−¿_{j , it} M_it−1 . β^¿_{j, it}

U X_it=

∑

j =1 nit

β⁺_{j ,it}⋅Δm⁺_{j ,it} M_it−1 +

∑

j=1 nit

¿

(10)

Given that β⁺_j and −¿

β^¿_j are assumed to be symmetric random variables, we can express them as

3 We assume this independence for simplicity (it preserves the linearity of the model) and to maintain comparability with the Basu and BKN models.

β⁺_{j, it}= β⁺+ υ_{j , it}⁺ β^−¿+ υ^−¿,_{j , it}

¿

−¿=¿

β^¿_{j, it}

(11)

where υ⁺_{j ,it} and −¿

υ_{j ,it}^¿ are deviations from the expectation of β⁺_j and −¿

β^¿_j , respectively⁴. The means of these two variables are equal to 0 and are independent of the

other random variables – particularly the variables _{Δ m}

j ,it

+ and −¿

Δm_{j ,it}^¿ .

Substituting (11) in (10), we obtain

−¿+ξ_it, β^−¿⋅ U R_it^¿ U X_it

M_it−1= β⁺⋅ U R_it⁺+¿

(12)

being

−¿

Δm^−¿_{j ,it} M_it−1 ⋅υ^¿_{j ,it} Δm⁺_{j ,it}

M_it−1 ⋅ υj , it+ +¿

¿ ξ_it=

∑

j =1 nit

¿

, a random variable with a mean of 0 that is independent of

U R_it⁺ and −¿

U R_it^¿ .

In this model, β⁺ captures the average proportion in which the unexpected good

news of the period is incorporated into the net income of that period, whereas β^−¿

¿

4 For simplicity, we are considering that +¿

β^¿_j ^and −¿

β^¿_j are stationary variables. Ball and Easton (2013) argue, however, that they change over time: As the economic gains or losses occurring at the beginning of the year have an entire year to be recognized in earnings, they are expected to have a higher coefficient than those gains or losses occurring at the end of the year, which will have less time to be recognized in that year. The incorporation of time-varying coefficients would increase the complexity of our model, but it would not affect to the final conclusions.

measures the same for unexpected bad news. Consequently, the difference between these two parameters measures the timeliness difference between good and bad news:

conditional conservatism.

How does the aggregation effect affect the BKN-model estimates?

The BKN model captures good-news timeliness with the coefficient b⁺ of equation (3), bad-news timeliness with the coefficient b⁻ of equation (3), and differential timeliness by the difference between b⁻ and b⁺. According to our theoretical model, the average

contemporaneous relationship between earnings and good news is captured by _β⁺ and

between earnings and bad news is β^−¿

¿ ; the difference between β^−¿

¿ and β⁺ indicates the existence of a difference in the timely recognition of gains and losses. For the BKN model to estimate correctly the contemporaneous relationship between news and accounting income, therefore, the coefficient b⁺ should be an unbiased estimator of _β⁺ ,

and the coefficient b⁻ should be an unbiased estimator of β^−¿

¿ .

BKN estimates of the influence of good news on accounting net income.

Proposition 1: The aggregation effect makes the BKN model produce a biased estimation of the influence of good news on accounting earnings, except under highly restrictive conditions.

To demonstrate this proposition, we rewrite equation (12) by adding and subtracting

β^+¿−¿⋅U R_it^¿

¿

, obtaining a version that resembles the BKN model:

β^−¿−β⁺

¿¿

−¿+ ξ_it.

−¿=β⁺⋅UR_it+¿

−¿+ ξ_it±β⁺⋅ U R_it^¿ β^−¿⋅ U Rit¿

U X_it

M_it−1= β⁺⋅ U R_it⁺+¿

(13)

Let ^b⁺ denote the OLS estimation of b⁺ from equation (3). The value of this ^b⁺ converges in probability in

p lim ^b⁺=

Cov

(

^{U XM}it−1^it

, U R_it

)

Var

(

^{U R}it

) |

^{U Rit≥0 , (14)}

where Cov(.) represents the covariance operator and Var(.) represents the variance

operator. If we substitute ^{U Xit} Mit−1

by its value from equation (13), we obtain

β^−¿−β⁺

¿¿

−¿+ ξ_it, UR_it

β⁺⋅UR_it+

(

^{⋅U Rit¿¿Var}

⁽

^{U R¿} it

) |

^{U Rit≥0=}

¿ β^−¿−β⁺

¿¿ UR_it, U R^¿_it −¿

Cov⁽¿ ¿ Var

(

^{U R}it

)

⁺

Cov

(

^URit, ξ_it

)

Var

(

^{U R}it

)

⏞

=0

|U R_it≥0=

β^−¿−β⁺

¿

−¿¿ UR_it, U R^¿_it

Cov⁽¿ ¿

Var

(

U R_it

)

^{|U Rit≥0.}

¿=

(

^β+

^|

^{U R}it≥0

)

^{+ θ}1

+, being θ₁⁺=¿ Cov¿

¿¿ p lim ^b⁺=¿

(15)

Consequently, ^b⁺ is equal to β⁺

|

UR_it≥0 plus a bias that we have noted by _θ

1 + . This bias arises because the estimation of the good-news timeliness in the BKN model ignores the influence of bad news on earnings for those observations with positive market returns, making it an omitted-variable bias. For ^b⁺ to be an unbiased estimator of

β⁺

|

URit≥0 , θ₁⁺ should be exactly equal to 0 – which will happen only if one of two conditions is met:

Condition 1: β^−¿= β⁺

|

U R_it≥0

¿

, which implies that there would be no

differentiation in the timely recognition of good and bad news and therefore no

accounting conservatism. Therefore, the BKN model would not be affected by this bias in the absence of conservatism.

Condition 2:

−¿,U R_it U R_it^¿

Cov⁽¿=0^|U R_it≥0

¿

. This condition implies that positive market

returns are independent of the bad news of the period.

That the omitted-variable bias θ₁⁺ is equal to 0 is a necessary condition for concluding that ^b⁺ is an unbiased estimator of the conditional relationship between good news and accounting income, given that unexpected returns are positive

(

^β+

^|

^{U R}it≥0

)

. But it is not a sufficient condition to conclude that ^b⁺ is an unbiased estimator of the unconditional relationship

(

β⁺

)

, unless the conditional and the

unconditional relationships are equal

(

^β+=

⁽

^β+

^|

^{U Rit≥0}

⁾ )

⁵. If they are different

(

^β+≠

⁽

^β+

^|

^{U R}it≥0

) )

^{, ^b+} would be affected by a second bias, which would be equal to the difference between the conditional and the unconditional coefficients. In summary:

p lim ^b⁺=

(

^β+

^|

^{U R}it≥0

)

^{+ θ}1

+= β⁺+

[ ⁽

^β+

^|

^{U Rit≥0}

⁾

^−β+

]

^{+ θ1+= β++ θ1++ θ2+}

being θ₂⁺=

[ ⁽

^β+

^|

^{U Rit≥0}

⁾

^−β+

]

^{. (16)}

The bias denoted by θ₂⁺ is a truncated-sample bias, because ^b⁺ is estimated using only those observations with positive unexpected returns – not observations with negative unexpected returns. Dietrich et al. (2007) also indicated that the Basu model could suffer from a truncated-sample bias that arises from sampling on an endogenous variable (market returns). Our justification for the truncated-sample bias is different: The

5 This would happen if the good-news timeliness conditioned on positive unexpected returns is equal to the good-news timeliness conditioned on negative unexpected returns

(

^β+

^|

^{U R}it≥0= β⁺

|

U R_it<0

)

^.

cause of the possible truncated-sample bias is the existence of different timeliness coefficients for the subsamples of positive and negative unexpected returns.

An implicit assumption in the BKN model, therefore, is that the unconditional contemporaneous relationship between good news and earnings is equal to the

conditional relationship, given that unexpected returns are positive. For this assumption to hold, good-news timeliness must remain unchanged for positive and negative values of unexpected stock returns. Although this assumption may seem reasonable, various researchers have argued that there can be factors other than accounting conservatism that could produce a nonlinear relationship between earnings and returns: the abandonment option (Beaver and Ryan, 2009; Hayn, 1995), earnings management (Watts, 2003), firm’s life cycle (Collins et al., 2014), or sticky costs (Banker et al., 2013), for example.

Additionally, Pope and Walker (1999) modeled the relationships among reported earnings, permanent earnings, and returns, and used this model for interpreting the regression coefficients of the Basu model⁶. According to their model, gains and losses timeliness coefficients are affected by conservatism, but they are also directly related to the cost of capital. The coefficient of the relationship between earnings and good news may therefore vary between the subsamples of positive and negative unexpected returns if there are differences in the cost of capital between those subsamples, even in absence of accounting conservatism. Finally, other papers have documented asymmetries in the return–earnings relationship that are unrelated to conditional conservatism: Hsu et al.

(2012) and Collins et al. (2014) have found that a substantial proportion of the differential

6 Pope and Walker (1999) define permanent earnings as a term in perpetuity, which, capitalized at the firm’s cost of capital, equals the stock price. Reported earnings are then equal to permanent earnings plus two addends. The first addend incorporates the under-recognition of good news and the over-recognition of bad news of the same period; the second addend incorporates the effect of prior period news on current period earnings.

timeliness coefficient in the Basu model is produced by the relationship between returns and cash flows; Ball and Easton (2013) differentiate between the recognition of expenses due to the matching principle (matched expenses) and the recognition of expenses due to changes in expectations regarding earnings of future periods (expectations element of expenses). They observe an asymmetric relationship between returns and the expectations element of expenses consistent with conditional conservatism, but their results also indicate the existence of asymmetric relationships between returns on the one hand and sales revenue and matched expenses on the other. Given that conditional conservatism does not justify the latter two asymmetric relationships, these results support the idea that the asymmetric returns–earnings relationship is not entirely attributable to conditional conservatism. In summary, all these works point to the existence of a nonlinear returns–

earnings relationship in the absence of conditional conservatism, which may produce a non-zero truncated-sample bias in the BKN-model estimates.

Finally, there is the possibility that ^+¿_^b^¿ could be an unbiased estimator of β^+¿

¿ if the two biases (the omitted-variable bias and the truncated-sample bias) offset each other. We consider this possibility highly unlikely, however, because of the differing nature of these two biases.

BKN estimates of the influence of bad news on accounting net income.

Proposition 2: The BKN model produces a biased estimation of the influence of bad news on accounting earnings, except under highly restrictive conditions.

To demonstrate this proposition, we can follow the same method we employed to demonstrate our first proposition. Thus, we rewrite equation (12) by adding and

subtracting ^{β−¿⋅ U Rit+}

¿ β^−¿−β⁺

¿¿ β^−¿⋅ UR_it−¿

β^−¿⋅ U R_it⁺=¿

−¿+ ξt±¿ β^−¿⋅ U R_it^¿ U X_it

M_it−1= β⁺⋅ U Rit++¿

(17)

The value of the OLS estimation of b⁻ −¿

^b^¿

¿

converges in probability in

−¿=

Cov

(

^{U XM}it−1^it

, U Rit

)

Var

(

^{U R}it

) |

^{U Rit<0 .}

p lim ^b^¿

(18)

Substituting ^{U Xit}

M_it−1 by its value from equation (17) and operating, we get

β^−¿−β⁺

¿

Cov

(

^{Varβ−¿⋅U R}

⁽

^{U Ritit−}

⁾

^{(¿⋅U Rit++ ξit, U Rit)}

|

^{U Rit<0=}

β^−¿−β⁺

¿¿ β^−¿

¿

−¿, β^−¿−β⁺

¿¿

−¿=−¿

¿U R_it≥0)+ θ₁^¿being θ₁^¿ β^−¿−¿

¿=¿

−¿=¿

¿ p lim ^b^¿

(19)

Therefore, ^−¿_^b^¿ measures the conditional timeliness of bad news β^−¿

¿U R¿_it<0⁾

¿

, with a

bias −¿

θ₁^¿

¿

, generated by the omission of the good news of those observations with

negative unexpected returns. As for Proposition 1, this bias could be equal to 0 in the absence of accounting conservatism or if the covariance between good news and unexpected returns for those observations with negative returns is 0.

Additionally, ^−¿_^b^¿ would be affected by a truncated-sample bias if the conditional timeliness and the unconditional timeliness of bad news differ

β^−¿

¿

β^−¿≠

(

^¿URit<0⁾

)

¿¿

– if the timeliness of bad news varies between those observations

with positive and negative unexpected returns β^−¿

¿¿ β^−¿

¿

¿¿

. In summary, we can conclude that

β^−¿

¿ β^−¿

¿ β^−¿

¿U R_it<0⁾−¿

¿ β^−¿ −¿

¿ β^−¿

¿U R_it<0⁾−¿

¿¿

−¿=¿

−¿+θ₂^¿being θ₂^¿ β^−¿+ θ₁^¿

−¿=¿

¿ β^−¿+¿

−¿=¿

¿U R_it<0)+ θ1¿

−¿=¿

¿ p lim ^b^¿

(20)

BKN estimates of the differential timeliness.

Proposition 3: The BKN model produces a biased estimation of the difference between the influence of bad news and good news on accounting earnings, except under highly restrictive conditions.

The BKN model captures the differential timeliness by the difference −¿−^b⁺

^b^¿ . The probability limit of that difference, according to equations (16) and (20), converges to

−¿−^b⁺

^b^¿

¿

−¿

−¿+θ₂^¿ β^−¿+ θ₁^¿−

(

^β++θ1

++ θ₂⁺

)

β^−¿−β⁺

¿¿ β^−¿

(

^β-

^|

^{U R}it≥0

)

^−¿

¿ β^−¿−β⁺

¿¿ U R_it, U R_it^¿ −¿

Cov⁽¿ ¿

Var

(

^{U R}it

)

^{|U Rit}≥0, and θ₂⁺=

[ ⁽

^β+

^|

^{U Rit≥0}

⁾

^−β+

]

^.

−¿=¿

−¿=−¿

¿being :θ₁^¿ p lim ¿ ¿

(21)

For ^{−¿− ^b}

+

^b^¿ to be an unbiased estimation of β^−¿−β⁺

¿ , the four biases should be equal to 0 (the conditions for each bias to be equal to 0 have been described

previously) or the sum of the four biases should be exactly equal to 0. Because there is no theoretical reason for expecting the four biases to sum to 0, we conclude that the

aggregation effect makes the BKN-model estimate of differential timeliness a biased

estimator of the real differential timeliness. The sign and magnitude of the bias is an empirical issue, however, and depends on the sum of the four biases.

IV. EMPIRICAL ESTIMATION OF THE AGGREGATION BIASES

To estimate empirically the sign and magnitude of the aggregation biases in the BKN model, we estimate the following empirical model, based on equation (12),

−¿+ ε_{1 it}, U X_it

M_it−1= α0+β0⋅ U Rit+

+β1⋅U Rit¿ (22)

where U R_it⁺ is the proxy for good news, −¿

U R_it^¿ the proxy for bad news of the period, α0 is the intercept of the model, coefficient β0 represents the average influence of good news, and β1 the average influence of bad news on the unexpected accounting net income

of the same period. β0 and β1 are then the estimates of the model for β⁺ and β^−¿

¿ . This model has been estimated using the full sample (unconditional estimates β0 and β1), the subsample of observations with positive unexpected returns (conditional estimates

β₀

|

^{U R}it≥0 and β₁

|

^{U R}it≥0 ), and the subsample of observations with negative unexpected returns (conditional estimates β0

|

^{U R}it<0 and β₁

|

^{U R}it<0 ).

Additionally, the omitted-variable biases ( −¿

θ₁^¿ and θ₁⁺ ) depend on the ratio of the conditional covariance of unexpected returns with good (bad) news on the variance of unexpected returns, given that unexpected returns are positive (negative). Given that those conditional ratios are equal to the conditional regression coefficients of good (bad) news on unexpected returns, we estimate the following regression model to compute them:

U R⁺_it= γ₀+ γ₁⋅ UR_it+ε_{2 it}

|

^{U R}it<0, being γ₁=Cov

(

^{U R}it +, U R_it

)

Var

(

^{U R}it

) |

^{U Rit<0}

−¿= γ₂+ γ₃⋅ UR_it+ε_{3 it} U R_it^¿

¿

−¿, U R_it U R_it^¿

¿ Cov⁽¿ ¿

Var

(

^{U R}it

)

^{|U Rit≥0.}

¿

(23)

Using the estimates from expressions (22) and (23), we estimate the empirical values of the biases presented on equation (21) in the following form:

−¿=−

[ ⁽

^β1

^|

^{U Rit<0}

⁾

⁻

⁽

^β0

^|

^{U Rit<0}

⁾ ]

^{⋅ γ1}

^θ₁^¿

−¿=

[ ⁽

^β1

^|

^{U Rit≥0}

⁾

^−β1

]

^θ₂^¿

^θ₁⁺=

[ ⁽

^β1

^|

^{U Rit≥0}

⁾

⁻

⁽

^β0

^|

^{U Rit≥0}

⁾ ]

^{⋅ γ3}

^θ₂⁺=

[ ⁽

^β0

^|

^{U Rit≥0}

⁾

^−β0

]

^.

(24)

Sample and data.

The empirical tests in this paper rely on a sample comprising all the nonfinancial firms listed on Standard and Poor’s Compustat between 1962 and 2009, with non-missing values for the variables net income and book value. Market data were obtained from the Center for Research in Security Prices (CRSP) data file. Those observations with missing data for return variables or with a beginning-of-period price of common share under 1 USD were also removed from the sample. We also dropped the top and bottom 1% of observations for all the continuous variables, in order to eliminate the influence of outliers. The final sample comprises 97,161 observations, corresponding to 8,890 firms.

Variables.

The dependent variable is unexpected accounting net income (UXit), computed as net income before extraordinary items (Xit) minus the expected net income before

extraordinary items (E[Xit]) and normalized by the market value of equity at the

beginning of the period (Mit−1). To compute the expected net income, we have used the average of the three measures proposed by Ball et al. (2013b)⁷.

The unexpected-returns variables

−¿

U R_it, U R_it⁺, and U R_it^¿

¿

were also estimated

following Ball et al. (2013b). To calculate UR , we computed the composite stock returns^it (Rit) for each observation over the 12 months ending 3 months after the fiscal year. We then divided the observations for each year among 5x5 portfolios, by sorting the observations on market capitalization and book-to-market ratio. We used the average return of each portfolio as the measure of expected stock returns for the firms of that portfolio and computed unexpected returns as the difference between total returns and expected stock returns⁸.

−¿

U R_it⁺ and U R_it^¿ represent, respectively, the accumulated sum of the good news and bad news of the year. To estimate these variables, we divided the year into intra-year intervals and computed the unexpected cum-dividend stock-price variations for

7 Ball et al. (2013b) suggest three approaches to computing the expected net income: the lagged net income, the times-series average of net income, and the expectation from a model estimated at the industry-year level. For the sake of simplicity, we have used a single measure computed as the average of the former three. We have repeated our analysis using each of the three measures of expected net income separately.

The results (not tabulated) do not differ qualitatively from those reported.

8Various authors have used the market-adjusted stock returns as the returns variable. We have also estimated our models using market-adjusted stock returns, but the results (not tabulated) are qualitatively identical to those reported for the non-adjusted stock returns.

each interval. U R_it⁺ −¿

U R_it^¿

¿

was then computed as the sum of all the positive

(negative) unexpected stock price variations in a year, divided by the stock price at the beginning of that year.

The validity of these good- and bad-news proxies will be affected by the length of the intra-year periods used for their computation. On the one hand, one could argue that the shorter the period, the lower the probability of aggregating good and bad news in these intra-year market returns and the more robust the proxies are to aggregation bias The use of periods that are too short presents some disadvantages as well – the possibility that a single event can affect returns for a longer period (price drift), for example, or that the short-period returns can be simple noisy variations, produced by market

microstructure effects rather than economic causes. Consequently, the measure of the good and bad news of the period implies a tradeoff between robustness to the aggregation effect (shorter periods increase robustness) and the problems of short-term market-value variations (longer periods reduce these problems).

In this study, we have estimated −¿

U R_it⁺ and U R_it^¿ using quarterly

(

^{UR _ q}it +

and

UR _ q−¿ _it^¿

¿

, monthly

(

^{UR _ m}it+

and

UR _ m−¿ _it^¿

¿

, and daily stock returns

(

^{UR _ d}it+ and UR _ d^−¿ _it^¿

¿ .

UR _ q−¿ _it^¿ UR _ q_it⁺¿

is computed as the sum of all the quarterly

good (bad) news divided by the stock price at the beginning of the period. To estimate quarterly good (bad) news, we have calculated the difference between each quarterly

cum-dividends variation in the stock price and the quarterly expected variation in the stock price, computed as the product of the stock price at the beginning of the year and the expected market return for that company and year, divided by 4. Positive differences are classified as good news and negative differences as bad news. For computing

UR _ m−¿ _it^¿ UR _ m_it⁺¿

and

UR _ d−¿ _it^¿ UR _ d_it⁺¿

, we have followed the same approach, using monthly and

daily variations in prices and the monthly and daily expected return⁹. Results.

Descriptive statistics.

Table 1 reports the descriptive statistics of the variables in this study. Panel A of Table 1 shows the mean, standard deviation, and percentiles of the variables for the full sample.

Panel B reports those statistics for the subsamples of observations with positive values of unexpected returns and Panel C reports them for the negative values.

For the full sample, consistent with previous research (Ball et al., 2013b; Patatoukas and Thomas, 2011), earnings and unexpected earnings both show a left-skewed distribution, whereas, returns and unexpected returns show a right-skewed distribution. The

decomposition of unexpected returns into good news and bad news proxies shows that the right-skewness of unexpected returns is also observed for the good news proxy – but not for the bad news proxy, which exhibits a left-skewed distribution. The skewness of

9 We want to highlight the fact that, by calculating our good and bad news proxies using this method, we

obtain

−¿.

−¿= UR _ d_it⁺+ UR _ d_it^¿

−¿= UR _ m_it⁺+ UR _ m_it^¿ U R_it= UR _ q_it⁺+ UR _ q_it^¿

earnings and the skewness of good news and bad news proxies remain unchanged when we divide the full sample into the subsamples of positive and negative unexpected

returns. Unexpected earnings, returns, and unexpected returns, however, present different skewness for the two subsamples: They are right-skewed for observations with positive unexpected returns and left-skewed for observations with negative unexpected returns.

INSERT TABLE 1 ABOUT HERE

Table 2 reports correlation coefficients for each pair of variables. Following the design used in Table 1, the coefficients are reported in three panels: Panel A displays the

coefficients for the full sample, and Panels B and C present the correlation coefficients for the subsamples formed by the positive and negative observations of the unexpected- returns variable, respectively. The reported correlations for the full sample resemble those reported in similar papers, such as those of Patatoukas and Thomas (2013) and Ball et al.

(2013b). Two key correlations, which determine the sign of the omitted variable biases

^θ₁⁺ and −¿

^θ₁^¿ , are those between unexpected returns and bad news conditioned on positive values of unexpected returns and between unexpected returns and good news conditioned on negative values of unexpected returns. The Pearson coefficient for the first is significantly negative for the three bad news proxies (−0.0450, −0.2409 and −0.4672), so the sign of the omitted variable bias for the relationship between earnings and good news would also be negative ( θ1+

<0 ). The correlation between unexpected returns and good news conditioned on the negative values of unexpected returns, however, is positive for the quarterly and monthly based proxies but negative for the daily based proxy (0.2216,

0.1221 and −0.1350), which would produce a negative omitted variable bias in the case of

quarterly and monthly based proxies ( −¿<0

θ₁^¿ ), and a positive omitted variable bias for

the daily-based proxy ( −¿>0 θ₁^¿ ).

INSERT TABLE 2 ABOUT HERE

Empirical estimation of the influence of the aggregation effect on the BKN model.

Table 3 reports the results of the pooled OLS estimation of the BKN model and the estimates of the models necessary to compute the aggregation biases according to the expression (24). Panel A of Table 3 shows the estimates of the BKN. These results indicate a good-news timeliness of 0.0332 and a bad-news timeliness of 0.1185, with the differential timeliness being 0.0853. These results are similar to those reported by BKN or Patatoukas and Thomas (2013) for the relationship between unexpected returns and unexpected accounting income. Panel B reports the estimates of the model (22)using the full sample and differentiating between the level of aggregation used to compute the good- and bad-news proxies. Comparing these results with the estimates of the BKN model, it can be seen that the estimates of the good-news timeliness (coefficient β0) are higher than those observed for the BKN model, the bad-news timeliness estimates are lower (β1< b0 + b1), and the differential timeliness estimates are also lower (β1 ̶ β0< b1).

Furthermore, as we reduce the duration of the period to compute intra-year gains, the differences between the two models are wider, which would be compatible with the

existence of an aggregation bias. The estimates of model (22) also show that the magnitude of differential timeliness decreases as the time period used to compute the intra-year news narrows, moving from 0.0274 for quarterly unexpected returns to 0.0157 for monthly, and only 0.0031 for daily. Although the difference is still positive, its magnitude is considerably lower than the BKN estimate of differential timeliness, which indicates that the BKN estimate of conditional conservatism is considerably

overestimated.

A comparison between the estimates of the BKN model and the unconditional estimates of model (22) provides an idea about the total bias produced by the aggregation effect. To differentiate between the omitted-variable and the truncated-sample bias, we computed the estimates of model (22) for the subsamples of positive and negative values of unexpected returns.

Panel C reports the estimates of model (22), using only the subsample of observations with positive unexpected returns. The estimates for good-news timeliness

(

^β0

|

^URit≥0

)

are 0.0338 for the quarterly levels of aggregation, 0.0340 for the monthly levels, and 0.0323 for the daily levels. All these estimates are close to b0, indicating that the coefficient b0 does not suffer from an omitted-variable bias of significant magnitude.

The estimates also show that bad-news timeliness

(

^β1

|

^{U R}it≥0

)

is similar to good-news timeliness for this subsample, making the differential timeliness close to 0, which is not significantly different from 0 for the daily based news¹⁰.

Panel D reports the estimates of the model using the subsample of observations with negative unexpected returns. The estimates for the bad-news timeliness coefficients

10 The fact that the differential timeliness estimates are so close to 0 justifies the small magnitude of the omitted-variable bias, for this bias depends on that differential timeliness.

(

^β1

|

^{U R}it<0

)

are 0.1140, 0.1178, and 0.1154, respectively. These estimates are close to the BKN-model estimate (0.1185), indicating that the omitted-variable bias will not be highly relevant for this parameter either. As for the estimates for the subsample with positive unexpected returns, however, good-news and bad-news timeliness are of similar magnitudes, producing a differential timeliness close to 0. Only the daily based estimate differs significantly from 0.

A comparison of the estimates reported in Panel C and Panel D reveals that the coefficients for those two subsamples are not equal. Specifically, timeliness coefficients for both good and bad news are smaller for the positive unexpected-returns subsample than for the negative unexpected-returns subsample. Thus, the results indicate that both good news and bad news are recorded in a more timely fashion when the unexpected returns are negative than when unexpected returns are positive, but that the difference in the timely recognition of gains and losses does not change significantly between positive and negative unexpected returns¹¹, thereby demonstrating the existence of asymmetry in the earnings–returns relationship that is not produced by conditional conservatism. This asymmetry produces an essential truncated-sample bias in the BKN-model estimates.

Thus, the BKN model underestimates good-news timeliness, because it accounts only for the relationship between good news and net income for those observations with positive unexpected returns (which is weaker) – not the same relationship for those observations with negative unexpected returns (which is stronger). It also overestimates the

relationship between net income and bad news because its estimate is based on

11 We tested the difference in the differential timeliness coefficients observed for the two subsamples to determine if they were significantly different from 0. The results indicated that the difference was not significantly different from 0 when the quarterly and monthly based variables are used but significantly different from 0 for the daily based variables.

observations with negative unexpected returns (stronger relationship) – not on

observations with positive unexpected returns (weaker relationship). As a result of the underestimation of good-news timeliness and the overestimation of bad-news timeliness, the differential timeliness is also overestimated.

Finally, Panel E of Table 3 reports the estimates of the model (23), which are necessary to estimate the omitted-variable biases.

INSERT TABLE 3 ABOUT HERE

Table 4 reports the estimates of the aggregation biases, as indicated in equation (24), using the coefficients reported in Table 3. Panel A reports the results obtained using the quarterly based news variables. The first bias reported is the omitted-variable bias. As we have discussed, these biases are expected to be of low magnitude, given the low

differential timeliness in the conditional estimations of model (22). It can be seen that the omitted-variable biases are very close to 0 and that none of them are significant.

Regarding the truncated-sample bias, however, the results show that the BKN estimate for good news (0.0332) is affected by a significantly negative truncated-sample bias of

−0.0200 and the BKN estimate for bad news (0.1185) by a positive bias of 0.0380. Thus the differential timeliness (0.0853) is overestimated by 0.0580.

The results obtained with the monthly (Panel B) and daily (Panel C) based news variables are similar to those observed with the quarterly based variables: The omitted- variable bias is of small magnitude and usually not significant. All three truncated-sample biases are significant, however, and have the same signs as do the quarterly based

variables. The magnitude of these biases is greater as we reduce the duration of the period, indicating that shorter periods are less affected by these biases.

INSERT TABLE 4 ABOUT HERE

In summary, the results indicate that the BKN-model estimates are severely affected by the aggregation biases: It underestimates good-news timeliness, overestimates bad-news timeliness and, consequently, overestimates differential timeliness. Furthermore, our results demonstrate the existence of asymmetry in the returns–earnings relationship that is not caused by conservatism: The conditional timeliness of both good news and bad news estimated using the subsample of negative unexpected returns is higher than the

conditional timeliness estimated on the subsample of positive returns. But differential timeliness remains almost equal for the two subsamples.

Finally, when we use different proxies for good and bad news, the results show a significantly positive differential timeliness, which would support the existence of conditional conservatism, but its magnitude is considerably smaller than previously reported in the extant literature – practically equal to 0 when it is estimated using daily based news measures.

Empirical estimation of the influence of the aggregation effect on the Basu model.

Although the BKN model is free from the biases that arise from the correlation between expectations of returns and net income or from the correlation of one of those

expectations with the unexpected portion of the other variable (Ball et al., 2013b;

Patatoukas and Thomas, 2013), it also suffers from some limitations. Thus, its

effectiveness in addressing the former biases depends upon the estimates of unexpected returns and unexpected earnings.

Ball et al. (2013b) highlight the difficulty of estimating expected returns over a long horizon, and argue that the error in its estimation will prevent the complete elimination of the bias. Additionally, Patatoukas and Thomas (2013) indicate that the BKN model is less easily linked to theory than the original Basu model is. Finally, most of the studies that have measured conditional conservatism with differential timeliness have relied on the original Basu model, using net income as a dependent variable and raw returns as an independent variable. Consequently, we have also estimated the aggregation effect of biases on the original Basu model. To perform this analysis, we have estimated the original Basu model (2) and modified versions of models (22) and (23), in which unexpected earnings and unexpected returns have been substituted by total earnings and raw returns.

The results of the analysis for the original Basu model¹² depict a pattern similar to that observed for the BKN model: Basu model estimates good news timeliness with a negative bias and bad news timeliness with a positive bias, and differential timeliness is therefore overestimated as well. The decomposition of the total aggregation bias into the omitted-variable and the truncated-sample biases shows that the omitted-variable biases are greater and significant for the original Basu model than for the BKN model. The truncated-sample biases continue to be of greater magnitude than the omitted-variable biases are, however.

Time-series and cross-sectional variation of the aggregation bias.

12 Although these results are not tabulated, they are available from the authors.